Duality Dual problem Duality Theorem Complementary Slackness Economic
Duality • • Dual problem Duality Theorem Complementary Slackness Economic interpretation of dual variables
Primal Problem max +4 x 1 +1 x 2 +5 x 3 +3 x 4 s. t. +1 x 1 -1 x 2 -1 x 3 +3 x 4 ≤ 1 +5 x 1 +1 x 2 +3 x 3 +8 x 4 ≤ 55 -1 x 1 +2 x 2 +3 x 3 -5 x 4 ≤ 3
Vector View max +4 x 1 +1 x 2 +5 x 3 +3 x 4 s. t. +1 -1 -1 +3 +5 x 1 +1 x 2 +3 x 3 +8 x 4 -1 +2 +3 -5 1 ≤ 55 3
Linearly combine rows max +4 x 1 +1 x 2 +5 x 3 +3 x 4 s. t. +1 -1 -1 +3 +5 x 1 +1 x 2 +3 x 3 +8 x 4 -1 +2 +3 -5 1 ≤ y 1 55 y 2 3 y 3
Move y into vectors max +4 x 1 +1 x 2 +5 x 3 +3 x 4 s. t. +1 y 1 -1 y 1 +3 y 1 1 y 1 +5 y 2 x 1 +1 y 2 x 2 +3 y 2 x 3 +8 y 2 x 4 ≤ 55 y 2 -1 y 3 3 y 3 +2 y 3 +3 y 3 -5 y 3
max +4 s. t. ≥ ≥ Column Constraints +1 +5 +3 +1 y 1 -1 y 1 +3 y 1 +5 y 2 +1 y 2 +3 y 2 +8 y 2 -1 y 3 +2 y 3 +3 y 3 -5 y 3 ≤ 1 y 1 + 55 y 2 + 3 y 3
Dual Problem min +1 y 1 +55 y 2 +5 y 3 s. t. +1 y 1 +5 y 2 -1 y 3 ≥ 4 -1 y 1 +1 y 2 +2 y 3 ≥ 1 -1 y 1 +3 y 2 +3 y 3 ≥ 5 +3 y 1 +8 y 2 -5 y 3 ≥ 3
Observations • The objective value of any feasible primal solution is less than the objective value of any feasible dual solution • Duality Theorem – If both problems have an optimal solution, they are equal in value • Optimal dual solution can be read off of final dictionary • Dual solution serves as a certificate of optimality – Quick verification of optimality of primal solution
Example from Text Maximize s. t. 4 x 1 5 x 1 -x 1 + x 2 - x 2 + 2 x 2 Final Dictionary x 2 = 14 - 2 x 1 - 4 x 3 x 4 = 5 - x 1 - x 3 x 6 = 1 + 5 x 1 + 9 x 3 z = 29 - x 1 - 2 x 3 + 5 x 3 + 3 x 4 - x 3 + 3 x 4 ≤ 1 + 3 x 3 + 8 x 4 ≤ 55 + 3 x 3 - 5 x 4 ≤ 3 - 5 x 5 - 2 x 5 +21 x 5 -11 x 5 + 3 x 7 - x 7 +11 x 7 - 6 x 7
Reading optimal solution to dual problem Final Dictionary x 2 = 14 - 2 x 1 - 4 x 3 x 4 = 5 - x 1 - x 3 x 6 = 1 + 5 x 1 + 9 x 3 z = 29 - x 1 - 2 x 3 - 5 x 5 - 2 x 5 +21 x 5 -11 x 5 + 3 x 7 - x 7 +11 x 7 - 0 x 6 -6 x 7 Dual objective: min y 1 + 55 y 2 + 3 y 3 Linking slack variables with dual variables → x 6 associated with y 2 → x 7 associated with y 3 → x 5 associated with y 1 = 11 y 2 = 0 y 3 = 6
Linking back to original problem z = 29 - 1 x 1 - 2 x 3 -11 x 5 - 0 x 6 -6 x 7 Dual objective: min y 1 + 55 y 2 + 3 y 3 y 1 = 11 y 2 = 0 y 3 = 6 Original Primal Problem Maximize 4 x 1 + x 2 + 5 x 3 + 3 x 4 s. t. x 1 - x 2 - x 3 + 3 x 4 ≤ 1 5 x 1 + x 2 + 3 x 3 + 8 x 4 ≤ 55 -x 1 + 2 x 2 + 3 x 3 - 5 x 4 ≤ 3
Complementary Slackness 1 max +4 x 1 +1 x 2 +5 x 3 +3 x 4 s. t. +1 y 1 -1 y 1 +3 y 1 1 y 1 +5 y 2 x 1 +1 y 2 x 2 +3 y 2 x 3 +8 y 2 x 4 ≤ 55 y 2 -1 y 3 3 y 3 +2 y 3 +3 y 3 +1 y 1 + 5 y 2 – 1 y 3 = 4 -5 y 3 OR x 1 = 0
Complementary Slackness 2 max +4 x 1 +1 x 2 +5 x 3 +3 x 4 s. t. +1 y 1 -1 y 1 +3 y 1 1 y 1 +5 y 2 x 1 +1 y 2 x 2 +3 y 2 x 3 +8 y 2 x 4 ≤ 55 y 2 -1 y 3 3 y 3 +2 y 3 +3 y 3 -5 y 3 +1 x 1 - x 2 – x 3 +3 x 4 = 1 OR y 1 = 0
Checking Optimality Without Certificate • Given candidate primal solution x • Write down equation in dual variables y – Use dual constraints at equality corresponding to components of x ≠ 0 – Add in equation yj = 0 if primal constraint i is not tight – If solution is a nondegenerate basic solution, there is a unique solution y
Example from Text Maximize s. t. Candidate x 1 = 0, System of 1 = -y 1 + 3 = 3 y 1 + 0 = 4 x 1 + x 2 + x 1 - x 2 5 x 1 + x 2 + -x 1 + 2 x 2 + solution x 2 = 14, x 3 = constraints y 2 + 2 y 3 8 y 2 - 5 y 3 y 2 5 x 3 3 x 3 + + + - 3 x 4 ≤ 1 8 x 4 ≤ 55 5 x 4 ≤ 3 0, x 4 = 5
Economic interpretation of dual variables • Values of optimal dual variables (yi) give the marginal value of small increases or decreases of the given resource (bi) – Requires optimal basic solution to be a nondegenerate basic optimal solution – See worksheet for examples
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